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566 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61,NO. 2, FEBRUARY 2012
Multihop Wireless Relaying Systems in the Presence
of Cochannel Interferences: Performance
Analysis and Design Optimization
Salama Said Ikki, Member, IEEE, and Sonia Aïssa, Senior Member, IEEE
Abstract—A study of the effect of cochannel interference on
the performance of multihop wireless networks with amplify-and-
forward (AF) relaying is presented. Considering that transmis-
sions are performed over Rayleigh fading channels, first, the exact
end-to-end signal-to-interference-plus-noise ratio (SINR) of the
communication system is formulated and upper bounded. Then,
the cumulative distribution function (cdf) and probability density
function (pdf) of the upper bounded end-to-end SINR are deter-
mined. Based on those results, closed-form expression for the error
probability is obtained. Furthermore, an approximate expression
for the pdf of the instantaneous end-to-end SINR is derived, and
based on this, a simple and general asymptotic expression for
the error probability is presented and discussed. In addition, the
ergodic capacity of multihop wireless networks in the presence of
external interference is studied. Moreover, analytical comparisons
between AF and decode-and-forward (DP) multihop in terms of
error probability and ergodic capacity are presented. Finally,
optimization of the power allocation at the network’s transmit
nodes and the positioning of the relays are addressed.
Index Terms—Amplify-and-forward (AF), cochannel interfer-
ence, decode-and-forward (DF), ergodic capacity, error probabil-
ity, multihop wireless networks, optimization, power allocation,
relay positioning.
I. INTRODUCTION
MULTIHOP wireless networks, which have been an im-
portant field of research recently (see, e.g., [1] and [2]
and references therein) have a number of advantages over
traditional communication networks in terms of deployment,
connectivity, and capacity while minimizing the need for fixed
infrastructures. Relaying techniques enable network connec-
tivity where traditional architectures are impractical due to
location constraints and can be applied to cellular wireless local
area and hybrid networks.
The performance of multihop wireless networks over fading
channels has been extensively evaluated in terms of outage
Manuscript received May 11, 2011; revised October 2, 2011; accepted
November 18, 2011. Date of publication December 15, 2011; date of current
version February 21, 2012. This work was supported by a Discovery Acceler-
ator Supplement Grant from the Natural Sciences and Engineering Research
Council of Canada. This paper was presented in part at the IEEE Global
Communications Conference, Houston, TX, December 5–9, 2011. The review
of this paper was coordinated by Prof. T. Tsiftsis.
The authors are with the National Institute of Scientific Research—Energy,
Materials, and Telecommunications, University of Quebec, Montreal, QC H5A
1K6, Canada (e-mail: ikki@emt.inrs.ca; aissa@emt.inrs.ca).
Digital Object Identifier 10.1109/TVT.2011.2179818
probability, error probability, and ergodic capacity [1], [3]–[6].
In fact, to the best of our knowledge, existing research on
multihop wireless networks has focused on the transmission
itself in the network, with interference being ignored. Never-
theless, cochannel interference is an important issue. Consider-
ation of cochannel interference is indeed necessary because of
the aggressive reuse of frequency channels for high spectrum
utilization in different wireless systems, and multihop wireless
networks are no exception. Different from the aforementioned
research works, [7]–[10] and references therein aim to ana-
lytically study the impact of cochannel interference on the
performance of dual-hop systems. These works have assumed
interference affecting either the relay(s) or the destination(s) for
the special case of dual-hop transmission.
This paper provides a comprehensive analysis for the ef-
fect of cochannel interference in practical multihop wireless
networks, considering independent and nonidentical Rayleigh
fading interfering signals affecting the relays. Specifically,
using amplify-and-forward (AF) at the relays, the error and
outage probabilities are investigated assuming that a finite
number of cochannel interferers affect each relay node and the
destination node. Furthermore, analytical comparisons between
AF relaying and decode-and-forward (DF) relaying in terms
of error probability and ergodic capacity are provided and
discussed. Finally, to minimize the error probability, we investi-
gate three optimization schemes, namely, power allocation with
fixed locations for the relays, optimal relay positioning with
fixed power allocations, and joint optimization of the power
allocations and relays’ locations.
II. SYSTEM MODEL AND PRELIMINARIES
As shown in Fig. 1, we consider an N-hop wireless relaying
system in which a source node T0communicates with a destina-
tion node TNvia (N−1)half-duplex relays T1,T
2,...,T
N−1.
The medium-access control scheme allocates a frequency band
to the source for its transmission, which is further divided
into orthogonal subchannels across time using a time-division
scheme to permit half-duplex operation at the relays. It is
assumed that channel state information is only known at the
receive nodes. Therefore, equal portions of the total transmis-
sion time are allocated to each transmit node along the multihop
path. Furthermore, it is assumed that only one node transmits in
each time slot.
0018-9545/$31.00 © 2012 IEEE
IKKI AND AÏSSA: MULTIHOP WIRELESS RELAYING SYSTEMS IN THE PRESENCE OF COCHANNEL INTERFERENCES 567
Fig. 1. Multihop wireless network with cochannel interference.
In the nth time slot, the nth relay node Tnreceives a faded
noisy signal from the immediately preceding transmitting ter-
minal Tn−1and a finite number of faded cochannel interfering
signals from LTnexternal interferers. Relay Tnprocesses the
received signal and then forwards it to the next node, i.e., Tn+1,
in the next time slot. Thus, the signal received at the nth relay
node is given by
yn=En−1αnxn−1+EIn
LTn
j=1
βTn,j mTn,j +wn(1)
where αndenotes the fading gain of the channel between
terminals Tn−1and Tn, following flat Rayleigh fading distri-
bution; xn−1denotes the unit-energy signal transmitted from
the (n−1)th node; and En−1indicates the transmit energy
from the said node. In the second term of the right-hand-side
(RHS) of (1), mTn,j is the jth cochannel interferer’s signal
affecting the nth relay and assumed of unit energy, LTnis
the total number of interferers that affect the nth relay, EInis
the energy of the interference signal at the nth relay, and βTn,j
is the flat Rayleigh fading coefficient of the jth interference
channel. Finally, the third term of the RHS of (1), i.e., wn,
represents the additive white Gaussian noise (AWGN) term at
the nth relay, with zero mean and variance N0.
Assuming AF relaying, the nth relay amplifies its received
signal by a gain An[7]. In this technique, a relay cannot differ-
entiate between the signal originating from the preceding node
and the interference signals. In our system, the amplification
factor at the nth relay node is expressed as
An=1
En−1|αn|2+EInLTn
j=1 |βTn,j |2+N0
.(2)
Thus, the amplification process at the nth relay consists of
generating the signal xn=Anynand transmitting it to the next
node. Furthermore, we assume that all the fading coefficients
αn(n=1,...,N)and βTn,j (j=1,...,L
Tn;n=1...,N)
are mutually independent.
The input–output relation for the N-hop wireless network
with AF relaying in the presence of cochannel interference can
be written as
yN=αNEN−1
N−1
i=1
AiαiEix0
+
N−1
i=1
N−1
n=i
AnEnαn+1 ⎡
⎣EIi
LTi
j=1
βTi,j mTi,j ⎤
⎦
+EIN
LTN
j=1
βTN,j mTN,j
+
N−1
i=1
N−1
n=i
AnEnαn+1wi+wN.(3)
Using the mutual independence between αn(n=1,...,N)
and βTn,j (j=1,...,L
Tn;n=1,...,N), the effective end-
to-end signal-to-interference-plus-noise ratio (SINR) at the des-
tination node can be obtained in (4), shown at the bottom of the
page. Then, with the help of the method used in [3], the end-to-
end SINR given in (4) can be simplified to
γend =N
n=1 1+1+LTn
j=1 γIn,j
γn−1−1
(5)
where γn=|αn|2En−1/N0is the useful SNR at the nth relay,
and γIn,j =(EIn/N0)|βTn,j |2is the interference-to-noise ratio
(INR) at the said relay with respect to the jth interferer (j=
1,...,L
Tn).TheN-hop end-to-end SINR expression in (5) is a
generalization of the SINR given in [11] for the simple dual-hop
case. Furthermore, in the absence of cochannel interference,
i.e., when LTn=0forn=1,...,N, (5) reduces to the SNR
derived in [3].
III. PERFORMANCE ANALYSIS
A. CDF
To simplify the performance analysis, (5) should be ex-
pressed in a more mathematically tractable form. To achieve
this, a tight upper bound is proposed, in which the end-to-end
SINR, i.e., γend, is written as [11]
γend ≤γup = min γeff
1,γeff
2,...,γeff
N(6)
where γeff
n=γn/(1+LTn
j=1 γIn,j ). It is known that the cumu-
lative distribution function (cdf) of γup can be written as
Fγup (γ)=1−
N
n=1 1−Fγeff
n(γ)(7)
where Fγeff
n(γ)is the cdf of the effective SINR at the nth relay,
i.e., γeff
n. The cdf of γeff
ncan be derived as follows. Recall that
γend =|αN|2EN−1N−1
i=1 A2
i|αi|2Ei−1
N−1
i=1 N−1
n=iA2
nEn|αn+1|2N0+N0+N−1
i=1 N−1
n=iA2
nEn|αn+1|2EIiLTi
j=1 |βTi,j |2+EINLTN
j=1 |βTN,j |2
(4)
568 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61,NO. 2, FEBRUARY 2012
γeff
n=γn/(1+LTn
j=1 γIn,j ). In addition, γnand γIn,j are ex-
ponentially distributed random variables according to fγn(γ)=
(1/¯γ1)exp(−γ/¯γ1)and fγIn,j (γ)=(1/¯γIn)exp(−γ/¯γIn),
respectively, where ¯γn=E[|αn|2]En−1/N0is the average
SNR of the desired signal at the nth relay and ¯γIn=
E[|βTi,j |2]EIn/N0is the average INR at the said relay with
respect to each of the LTninterferers, and E[.]is the ex-
pectation operator. Hence, by using the approach in [12], the
cdf Fγeff
n(γ)can be written as Fγeff
n(γ)=1−((¯γn/¯γIn)/(γ+
¯γn/¯γIn))LTnexp(−γ/¯γn). Then, by substituting Fγeff
n(γ)into
(7), the cdf of γup can be expressed as
Fγup (γ)=1−exp −γ
N
n=1
1
¯γnN
n=1 ¯γn/¯γIn
γ+¯γn/¯γInLTn
.
(8)
Note that the outage probability, which is defined as Pout =
2Nr−1
0fγup (γ)dγ [3], corresponds to the cdf of γup evaluated
at 2Nr −1. Therefore, Pout can be written as Pout =Fγup
(2Nr −1), where Fγup is shown in (8).
B. Error Probability
For several constellations employed in practice, the system’s
error probability is of the form aQ(√bγ), where Q(x)
is the complementary error function defined as Q(x)=
(2/√π)∞
xexp(−x2)dx, and (a, b)are constants that depend
on the type of modulation. One way to evaluate the error
probability is the cdf-based approach used in [14]. Accordingly,
the average error probability, which is denoted PAF
b(e), can be
rewritten and directly expressed in terms of the cdf of γup as1
PAF
b(e)=ab
π
∞
0
exp(−bγ)
√γFγup (γ)dγ. (9)
To simplify the ensuing derivation, (8) should be expressed
in a more mathematically tractable form. For such, using partial
fraction, Fγup (γ)can be rewritten as
Fγup (γ)=1−exp −γ
N
n=1
1
¯γnN
n=1
LTn
i=1
λi,n
1+¯γIn
¯γnγi
(10)
where
λi,n =(¯γn/¯γIn)LTn−1
(LTn−1)!
×⎧
⎨
⎩
∂LTn−i
∂γLTn−i1+γ¯γIn
¯γnLTn1−Fγup (γ)
exp −γN
n=1
1
¯γn⎫
⎬
⎭γ=−¯γn
¯γIn
.
(11)
1In the presence of a large number of interfering signals, which is typical
in a wireless environment, the calculation of the average error is based on the
Gaussian assumption for the interference by applying the central limit theorem
(cf. [13] and references therein).
Then, substituting (10) into (9), the error probability can be
evaluated as
PAF
b(e)=ab
π⎡
⎢
⎣
∞
0
exp(−bγ)
√γdγ −
N
n=1
LTn
i=1
λi,n
×
∞
0
exp(−bγ)
√γ
exp −γN
n=1
1
¯γn
1+¯γIn
¯γnγidγ⎤
⎥
⎦.(12)
The expression in (12) can be solved as
PAF
b(e)=a1−√b
N
n=1
LTn
i=1
λi,n
ׯγn
¯γIn
Φ1
2;3
2−i;¯γn
¯γInb+
N
n=1
1
¯γn (13)
where Φ(x;y;z)is the confluent hypergeometric function of
the second kind [15, 9.211.4]. It should be noted that this
function is available in many popular software tools, such as
MATHEMATICA and MAPLE.
C. Approximate Analysis
Although the expression for Fγup (γ)in (8) enables numerical
evaluation of the system performance and may not be com-
putationally intensive, it does not offer insight into the effects
of the system parameters. We now aim at expressing Fγup (γ)
and PAF
b(e)in simpler forms. According to [16] and [17],
the asymptotic error probability can be derived based on the
behavior of the probability density function (pdf) of the output
SINR around the origin. By using Taylor’s series, Fγup(γ)given
in (8) can be rewritten as
Fγup (γ)→γ
N
n=1 #1
¯γn
(1+LTn¯γIn)$+H.O.T (14)
where H.O.T. stands for higher order terms. Hence, by using
(9), the asymptotic error probability can be written as
PAF
b(e)→a
2b
N
n=1 #1
¯γn
(1+LTn¯γIn)$+H.O.T (15)
where, as previously indicated, the values of aand bdepend
on the modulation scheme. Note that when LTn=0forn=
1,...,N, i.e., with no interference, (15) reduces to the well-
known formula provided in [17], which makes our results more
general. In addition, it can clearly be seen from (15) that the
error probability increases as LTnand ¯γInincrease and that it
decreases as ¯γnincreases.
D. Ergodic Capacity of AF Multihop Networks
Ergodic capacity, in the Shannon’s sense, is an important
performance metric since it quantifies the maximum achiev-
able transmission rate under which errors are recoverable. For
IKKI AND AÏSSA: MULTIHOP WIRELESS RELAYING SYSTEMS IN THE PRESENCE OF COCHANNEL INTERFERENCES 569
multihop networks with Nrelays operating according to the
AF protocol, the ergodic capacity can be expressed (in bits per
second per hertz) as
CAF =1
NE[log2(1+γup)] (16)
where, as indicated previously, γup is the upper-bound end-
to-end SINR. Unfortunately, the preceding expression cannot
be obtained in closed form. In the sequel, we derive an ap-
proximate expression for the ergodic capacity. Using Jensen’s
inequality, an upper bound on the ergodic capacity in (16) is
obtained as
CAF ≤1
Nlog2(1+E[γup]) (17)
where E[γup]is the expected value of the upper-bounded end-
to-end SINR at the destination node in the network. The latter
is obtained as
E[γup]=
∞
01−Fγup (γ)dγ
=
N
n=1
LTn
i=1
λi,n
∞
0
exp −γN
n=1
1
¯γn
1+ ¯γIn
¯γnγidγ (18)
which, after further manipulations, is solved to
E[γup]=
N
n=1
LTn
i=1
λi,n exp Ω¯γn
¯γInΩi−1¯γn
¯γIni
×Γ1−i, Ω¯γn
¯γIn(19)
where Ω=N
n=1(1/¯γn). Finally, substituting (19) into (17),
a closed-form expression for the upper bound on the ergodic
capacity can be obtained.
E. AF Versus DF
In a DF multihop network, the received signal at each relay
is fully decoded, reencoded, and then transmitted to the next
relay. To determine whether it is preferable to use multihop
analog relaying (AF) or perform digital regeneration (DF) of
the original signal in each stage of the relaying system, here,
we evaluate the end-to-end error probability and the ergodic
capacity of the multihop network when DF is implemented.
1) Error Probability: Let a bit be transmitted by the source
node, i.e., T0, to the destination node, i.e., TN, through N−1
regenerating relays, with each single hop characterized by error
probability Pb(n),givenn∈{1,2,...,N}. Then, the proba-
bility that the bit received at the destination node is different
from that transmitted by the source is given by [19]
PDF
b(e)=
N
n=1 ⎛
⎝Pb(n)
N
j=n+1
(1−2Pb(j))⎞
⎠.(20)
It should be noted that, for identical channels, PDF
b(e)can
be simplified to
PDF
b(e)=1
21−(1−2Pb(n))N.(21)
To determine PDF
b(e)given in (20) or (21), we need to find
the error probability per hop, i.e., Pb(n)for n=1,2,...,N.In
this vein, the per-hop error probability can be written as
Pb(n)=ab
π
∞
0
exp(−bγ)
√γFγeff
n(γ)dγ
=a1−b¯γn
¯γIn
Φ1
2;3
2−LTn;¯γn
¯γInb+1
¯γn.
(22)
Then substituting (22) into (20) (or in (21) for the case
with identical channels), a closed-from expression for the error
probability of DF multihop networks is obtained.
Furthermore, the approximate error probability in the mul-
tihop DF case can be obtained by finding an approximate
expression for the error probability of the individual links, i.e.,
Pb(n),n=1,...,N. It can be shown that the error probability
of the nth link can be approximated as
Pb(n)≈a
2b
1
¯γn
(1+LTn¯γIn).(23)
Substituting (23) into (20), the overall error probability can
be rewritten as
PDF
b(e)≈
N
n=1
a
2b
1
¯γn
(1+LTn¯γIn)
N
j=n+1
(1−2Pb(j))
)*+ ,
→1
(24)
≈
N
n=1
a
2b
1
¯γn
(1+LTn¯γIn).(25)
It is shown in (24) that the DF technique will slightly
outperform the AF technique in multihop networks given that
the product term, i.e., N
j=n+1(1−2Pb(j)), will have a value
less than 1.
2) Ergodic Capacity: Suppose that the transmitted code-
words from the source and the relays are chosen from a
Gaussian codebook. Then, the nth relay (n=1,...,N)can
decode the codeword transmitted from the (n−1)th relay with
achievable capacity of
Cn=1
NE[log2(1+γn)] .(26)
Using Jensen’s inequality, an upper bound on the ergodic
capacity in (26) is obtained as
Cn≤1
Nlog2(1+E[γn]) (27)
570 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2,FEBRUARY 2012
where Cndenotes the capacity of the nth link, and E[γn]can
be obtained as
E[γn]=exp(1/¯γIn)¯γn/¯γi
InΓ(1−LTn,1/¯γIn).(28)
Thus, the overall system capacity should be the minimum
of the capacity over each individual link. On the other hand,
according to the min-cut max-flow theorem [20], the overall
system capacity cannot be larger than the capacity of each link.
Therefore, the upper bound on the ergodic capacity in a DF
multihop network is given by
CDF = min(C1,C
2,...,C
N).(29)
Finally, the comparison between the ergodic capacity of
AF and DF multihop networks can be obtained as follows.
Applying Jensen’s inequality, one has
CDF >1
NE[min {log2(1+γ1),...,log2(1+γN)}]
=1
NE[log2(1+ min{γ1,...,γ
N})] ≥CAF (30)
which proves that a DF multihop network achieves a slightly
higher ergodic capacity than its AF counterpart.
IV. SYSTEM OPTIMIZATION FOR MULTIHOP
AMPLIFY-AND FORWARD WIRELESS NETWORKS
In our optimization study, we consider the scenario where
the received power decays with the inverse of the distance
(between a transmitter and a receiver) to the power of path-
loss exponent ηthat is greater than 1. Consider a distance D
between source T0and destination TN, and multihop trans-
mission through Nhops, with the nth-hop distance (between
nodes Tn−1and Tn) denoted dn. If the relays are positioned
along the line joining nodes T0and TNand are ordered in
sequence T1,T
2,...,T
N−1, then we have condition d1+d2+
···+dN=D. In the case of other kinds of positioning for the
relays, we have condition d1+d2+···+dN>D. Combin-
ing both conditions, we get N
n=1 dn≥D.
Now, since E[α2
n]=1/dη
nfor n=1,...,N, we can express
the average value of the nth SNR as ¯γn=(En−1/N0)d−η
n.
Therefore, the average error probability, which is given in (15),
can be rewritten as
PAF
b(e)= a
2b
N
n=1
N0
En−1
dη
n(1+LTn¯γIn).(31)
A. Adaptive Power Allocation Under Fixed Relays
Here, we derive the optimal power allocation that minimizes
the error probability subject to a sum-power constraint. Thus,
our optimization problem is formulated as
minimize PAF
b(e)= a
2b
N
n=1
N0
En−1
dη
n(1+LTn¯γIn)
subject to
N
n=1
En−1=ET,E
n−1>0,n=1,...,N (32)
where ETdenotes the total energy budget. The Lagrange cost
function can be written as
Υ= a
2b
N
n=1
N0
En−1
dη
n(1+LTn¯γIn)+φN
n=1
En−ET
(33)
where φis the Lagrange parameter. Upon setting the derivatives
of Υwith respect to En−1and φto zero, we get
∂Υ
∂En−1
=−a
2b
N0
E2
n−1
dη
n(1+LTn¯γIn)+φ=0,n=1,...,N
(34)
∂Υ
∂φ =
N
n=1
En−1−ET=0 (35)
and by solving the set of N+1 equations in (34) and (35),
we obtain a closed-form solution for the values of the optimum
power distribution, i.e., E∗
n−1,n=1,...,N, subject to sum-
power constraint
E∗
n−1=dη/2
n1+LTn¯γIn
N
n=1 dη/2
n1+LTn¯γIn
ET,n=1,...,N.
(36)
It is shown in (36) that, as the number of cochannel interfer-
ers that affect the nth relay increases, more energy will be given
to that node, and vice versa. This behavior can be expected
since as the nth link gets worse, the nth relay needs more energy
to compensate for the low quality of this link.
B. Relay Positioning Under Fixed Power Allocation
In this case, we assume that all nodes are aligned, i.e.,
d1+d2+···+dN=D. This assumption is reasonable since
the optimal positions of the relays must lie on the line joining
source node T0with destination node TNto minimize the
effect of the path loss. Under a predetermined power allocation,
the problem of finding the optimal position of the relays that
minimizes the system error probability can be stated as follows:
minimize PAF
b(e)= a
2b
N
n=1
N0
En−1
dη
n(1+LTn¯γIn)
subject to
N
n=1
dn=D, dn>0,n=1,...,N. (37)
The corresponding Lagrange cost function can be written as
Δ= a
2b
N
n=1
N0
En−1
dη
n(1+LTn¯γIn)+ψN
n=1
dn−D
(38)
where ψis the Lagrange parameter. Upon setting the derivatives
of Δwith respect to dnand ψto zero, we obtain
∂Δ
∂dn
=a
2b
N
n=1
N0
En−1
ηdη−1
n(1 + LTn¯γIn)+ψ=0
n=1,...,N (39)
IKKI AND AÏSSA: MULTIHOP WIRELESS RELAYING SYSTEMS IN THE PRESENCE OF COCHANNEL INTERFERENCES 571
∂Δ
∂ψ =
N
n=1
dn−D=0n=1,...,N. (40)
Then, solving the set of N+1 equations in (39) and (40)
yields a closed-form solution for the optimum relays’ posi-
tions, i.e.,
d∗
n=En−1
1+LTn¯γIn1
η−1
N
n=1 En−1
1+LTn¯γIn1
η−1
D, n =1,...,N. (41)
From (41), it is shown that as the number or strength of
the cochannel interferers that affect the nth relay increases
and/or as the energy that is assigned to the (n−1)th relay
node decreases, the nth relay should be closer to the preceding
node to compensate the reduction of the quality of this link, and
vice versa.
C. Joint Optimization of the Power Allocation
and the Relays’ Locations
The system performance can be further improved if the
power allocation and the relays’ positions are jointly optimized.
This optimization problem is formulated as
minimize PAF
b(e)= a
2b
N
n=1
N0
En−1
dη
n(1+LTn¯γIn)
subject to
N
n=1
dn=Dand
N
n=1
En−1=ET
with dn>0 and En−1>0forn=1,...,N. (42)
The corresponding Lagrange cost function can be written as
Ω= a
2b
N
n=1
N0
En−1
dη
n(1+LTn¯γIn)
+φN
n=1
En−ET+ψN
n=1
dn−D(43)
where ψand φare the Lagrange parameters. Then, setting the
derivatives of Ωwith respect to dn,En−1,φ, and ψto zero, and
solving the set of 2N+2 equations obtained, yields a closed-
form solution for optimum relays’ positions d∗
n(n=1,...,N)
and optimum power distribution E∗
n−1(n=1,...,N)
as follows:
d∗
n=1
1+LTn¯γIn1
η−2
N
n=1 1
1+LTn¯γIn1
η−2
D(44)
E∗
n−1=d∗
n
DET,n=1,...,N. (45)
Fig. 2. Average error probability of multihop AF relaying in the presence of
cochannel interference for different numbers of hops N(LTn=5and¯γIn=
0.001ET/N0,n=1,...,N).
D. Design Optimization for DF Multihop Networks
Recall that the focus of this paper is on multihop networks
implementing the AF protocol. However, as we did when we
performed the comparisons with the DF case in Section III-E,
we here provide a remark on the system optimization for the
DF case. Specifically, we show that minimizing the error prob-
ability in the case of AF multihop networks can be also used to
minimize the error probability of DF multihop networks. Note
that the error probability expression shown in (25) for the DF
case is the same as that given in (15) for the AF scenario. As a
result, all the expressions that have been previously proved in
the optimization of AF multihop networks can be directly used
for the optimization of DF multihop networks to determine the
power allocations and relay positions as per the three algorithms
detailed earlier.
V. I LLUSTRATIVE NUMERICAL RESULTS
Now, we provide numerical results for the error performance
assuming binary phase-shift keying modulation. Results are
plotted as a function of ET/N0, where ET=N
n=1 En−1,
and our analytical results are also compared with Monte Carlo
simulations.
In Fig. 2, it is assumed that the distance between the source
node and the destination node is normalized (D=1)and that
the distances between different terminals are identical (dn=
1/N ). The lower bound on the error probability is illustrated
along with the simulation results. As it is shown, in the low-
to-medium range of ET/N0, increasing ET/N0improves the
error probability performance because the dominant noise in
this region is the AWGN. On the other hand, at high ET/N 0,
error floors appear due to the cochannel interference, which
is independent of ET/N0. In addition, and as expected, the
error probability improves as the number of hops increases.
Furthermore, it is observed that the number of interfering
signals has no effect on performance in the low ET/N0range,
whereas degradation can be seen as ET/N0increases. Finally,
it is observed that the tightness of the proposed lower bound
decreases at low ET/N0values, particularly when Nincreases,
due to the fact that the accuracy of the SINR approximation in
(6) improves as ET/N0increases.
572 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 61, NO. 2,FEBRUARY 2012
Fig. 3. Comparison between multihop AF relaying and DF relaying in terms
of error probability in the presence of cochannel interference for different
numbers of hops N(LTn=5and¯γIn=0.001ET/N 0,n=1,...,N).
Fig. 4. Channel capacity of multihop AF relaying in the presence of cochan-
nel interference for different numbers of hops N(LTn=5and¯γIn=
0.001ET/N0,n=1,...,N).
Fig. 3 shows a comparison between AF and DF for multihop
networks with N=2 and 5 hops. It is clear from the figure
that DF multihop networks outperform their AF counterparts at
low values of ET/N0. Both types of networks are equivalent bit
error rate-wise at high ET/N0, and the difference between both
at low ET/N0is larger as the number of hops increases.
Fig. 4 shows the channel capacity for different numbers
of hops. As observed, increasing Ndoes not improve the
channel capacity, due to the fact that the number of orthogonal
channels needed increases as the number of hops increases, thus
decreasing the channel capacity by a factor of N. Finally, it
is important to mention that the gap between the exact results
and the ones corresponding to the equations derived is due to
Jensen’s property and not our developed bound.
Figs. 5 and 6 compare the system error performance for
different optimization schemes in dual-hop (N=2)and triple-
hop (N=3)networks, respectively. We can see that, with
the fixed relay positioning (dn=1/N ), adaptive power alloca-
tion outperforms the fixed power allocation scheme (En−1=
ET/N ). On the other hand, under fixed power allocation,
choosing the optimal location for the relays reduces the sys-
tem’s error probability significantly. When the power allocation
and relays’ locations are jointly optimized, the minimal error
probability is achieved. Furthermore, it is shown in the figures
that the optimal solution has no effect on the error performance
in the low ET/N0range. This is due to the fact that the most
Fig. 5. Average error probability for different adaptive algorithms in an AF
relaying network with N=2 hops in the presence of cochannel interference.
Fig. 6. Average error probability for different adaptive algorithms in an AF
relaying system with N=3 hops in the presence of cochannel interference.
dominant factor in this case is the AWGN, and it is known
that, for Rayleigh fading channels, the joint optimum solution
is a uniform power allocation with equal-distance placement
as the optimal position for the relays. This can be also proven
from (45). Furthermore, for high ET/N0, the error floor in case
N=3 and the joint optimization is used decreases by a 1.6
order of magnitude, compared the uniform power distribution
with equal-distance scheme.
VI. CONCLUSION
In this paper, we have studied the effects of cochannel inter-
ference on the performance of multihop wireless networks with
the AF relaying technique. The derived expression for the error
probability, based on a tight lower bound on the effective SINR,
provides a good match with the simulation results. Furthermore,
by using Jensen’s inequality, approximate expressions for the
ergodic capacity of multihop networks with AF or DF relay-
ing were obtained. In addition, simple high-SINR expressions
for the error probability were derived and compared for both
relaying schemes. Furthermore, we investigated the adaptive
power allocation scheme at the source and relay nodes under a
joint power constraint to minimize the system error probability
when the relays’ positions are fixed. We also found the optimal
positioning for the relays conditioned on fixed power allocation
and further studied joint optimization of the power allocations
and relays’ positioning. In addition, for a given total received
IKKI AND AÏSSA: MULTIHOP WIRELESS RELAYING SYSTEMS IN THE PRESENCE OF COCHANNEL INTERFERENCES 573
interference power, we proved that an equal power distribution
with equal-distance relay positioning yields the worst error
performance in the medium-to-high SNR range and the best
performance at low SNRs.
REFERENCES
[1] V. Asghari, A. Maaref, and S. Aıuml;ssa, “Symbol error probability analy-
sis for multihop relaying over Nakagami fading channels,” in Proc. IEEE
WCNC, Sydney, Australia, Apr. 2010, pp. 1–6.
[2] A. H. Forghani, S. S. Ikki, and S. Aıuml;ssa, “Novel approach for approxi-
mating the performance of multi-hop multi-branch relaying over Rayleigh
fading channels,” in Proc. IFIP WD Conf., Niagara Falls, ON, Canada,
Oct. 2011, pp. 1–3.
[3] M. O. Hasna and M. S. Alouini, “Outage probability of multi-hop trans-
mission over Nakagami fading channels,” IEEE Commun. Lett.,vol.7,
no. 5, pp. 216–218, May 2003.
[4] G. K. Karagiannidis, T. A. Tsiftsis, and R. K. Mallik, “Bounds of multihop
relayed communications in Nakagami-mfading,” IEEE Trans. Commun.,
vol. 54, no. 1, pp. 18–22, Jan. 2006.
[5] J. Boyer, D. D. Falconer, and H. Yanikomeroglu, “Multihop diversity
in wireless relaying channels,” IEEE Trans. Commun., vol. 52, no. 10,
pp. 1820–1830, Oct. 2004.
[6] G. Farhadi and N. C. Beaulieu, “On the ergodic capacity of multi-hop
wireless relaying systems,” IEEE Trans. Wireless Commun., vol. 8, no. 5,
pp. 2286–2291, May 2009.
[7] I. Krikidis, J. S. Thompson, S. McLaughlin, and N. Goertz, “Max–min
relay selection for legacy amplify-and-forward systems with interfer-
ence,” IEEE Trans. Wireless Commun., vol. 8, no. 6, pp. 3016–3027,
Jun. 2009.
[8] A. Agustin and J. Vidal, “Amplify-and-forward cooperation under
interference-limited spatial reuse of the relay slot,” IEEE Trans. Wireless
Commun., vol. 7, no. 5, pp. 1952–1962, May 2008.
[9] H. A. Suraweera, H. K. Garg, and A. Nallanathan, “Performance analysis
of two hop amplify-and-forward systems with interference at the relay,”
IEEE Commun. Lett., vol. 14, no. 8, pp. 692–694, Aug. 2010.
[10] S. S. Ikki and S. Aïssa, “Impact of imperfect channel estimation and co-
channel interference on dual-hop relaying systems,” IEEE Commun. Lett.,
2011, to be published.
[11] S. S. Ikki and S. Aïssa, “Performance analysis of dual-hop relaying sys-
tems in the presence of co-channel interference,” in Proc. IEEE GLOBE-
COM, Miami, FL, Dec. 2010, pp. 1–5.
[12] V. A. Aalo and J. Zhang, “Performance analysis of maximal ratio com-
bining in the presence of multiple equal-power co-channel interferers in
a Nakagami fading channel,” IEEE Trans. Veh. Technol., vol. 50, no. 5,
pp. 497–503, Mar. 2001.
[13] K. S. Ahn and R. W. Heath, “Performance analysis of maximum ratio
combining with imperfect channel estimation in the presence of cochannel
interferences,” IEEE Trans. Wireless Commun., vol. 8, no. 3, pp. 1080–
1085, Mar. 2009.
[14] Y. Chen and C. Tellambura, “Distribution function of selection com-
biner output in equally correlated Rayleigh, Rician, and Nakagami-m
fading channels,” IEEE Trans. Commun., vol. 52, no. 11, pp. 1948–1956,
Nov. 2004.
[15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and
Products., 7th ed. San Diego, CA: Academic, 2007.
[16] Z. Wang and G. B. Giannakis, “A simple and general parameterization
quantifying performance in fading channels,” IEEE Trans. Commun.,
vol. 51, no. 8, pp. 1389–1398, Aug. 2003.
[17] A. Ribeiro, X. Cai, and G. B. Giannakis, “Symbol error probabilities for
general cooperative links,” IEEE Trans. Wireless Commun., vol. 4, no. 3,
pp. 1264–1273, May 2005.
[18] A. Papoulis and S. U. Pillai, Random Variables and Stochastic Processes,
4th ed. New York: McGraw-Hill, 2002.
[19] E. Morgado, I. Mora-Jiménez, J. J. Vinagre, J. Ramos, and A. J. Caamano,
“End-to-end average BER in multihop wireless networks over fading
channels,” IEEE Trans. Wireless Commun., vol. 9, no. 8, pp. 2478–2487,
Aug. 2010.
[20] T. M. Cover and A. A. El Gammal, “Capacity theorems for relay chan-
nels,” IEEE Trans. Inf. Theory, vol. IT-25, no. 5, pp. 572–584, Sep. 1979.
Salama Said Ikki (M’09) received the B.S. degree in electrical engineering
from Al-Isra University, Amman, Jordan, in 1996, the M.Sc. degree in electrical
engineering from the Arab Academy for Science and Technology and Maritime
Transport, Alexandria, Egypt, in 2001, and the Ph.D. degree in electrical
engineering from Memorial University, St. John’s, NF, Canada, in 2008.
He is currently a Research Assistant with the Department of Electrical and
Computer Engineering, University of Waterloo, Waterloo, ON, Canada. His
research interests include communications and signal processing, specifically
wireless cooperative networks and multiple-input–multiple-output systems. His
current research interests include signal processing, imperfect estimation, and
cochannel interference for cooperative communications networks.
Sonia Aïssa (S’93–M’00–SM’03) received the Ph.D. degree in electrical and
computer engineering from McGill University, Montreal, QC, Canada, in 1998.
Since 1998, she has been with the National Institute of Scientific
Research—Energy, Materials, and Telecommunications (INRS-EMT), Uni-
versity of Quebec, Montreal, where she is currently a Full Professor. From
1996 to 1997, she was a Researcher with the Department of Electronics and
Communications, Kyoto University, Kyoto, Japan, and the Wireless Systems
Laboratories, NTT, Kanagawa, Japan. From 1998 to 2000, she was a Research
Associate with INRS-EMT. From 2000 to 2002, while she was an Assistant
Professor, she was a Principal Investigator in a major program of personal
and mobile communications of the Canadian Institute for Telecommunications
Research, leading research in radio resource management for code-division
multiple-access systems. From 2004 to 2007, she was an Adjunct Professor
with Concordia University, Montreal. In 2006, she was a Visiting Invited
Professor with the Graduate School of Informatics, Kyoto University. Her
research interests are in the area of wireless and mobile communications
and include radio resource management, cross-layer design and optimization,
design and analysis of multiple-input–multiple-output systems, cognitive and
cooperative transmission techniques, and performance evaluation, with a focus
on cellular, ad hoc, and cognitive radio networks.
Dr. Aïssa was the Founding Chair of the Montreal Chapter IEEE Women in
Engineering Society during 2004–2007, a Technical Program Cochair for the
Wireless Communications Symposium (WCS) of the 2006 IEEE International
Conference on Communications (ICC 2006), and the PHY/MAC Program
Chair for the 2007 IEEE Wireless Communications and Networking Con-
ference (WCNC 2007). She was also the Technical Program Leading Chair
for the WCS of IEEE ICC 2009 and a Cochair for the WCS of the IEEE
ICC 2011. She served as a Guest Editor for the EURASIP Journal on Wireless
Communications and Networking in 2006 and an Associate Editor for the
IEEE WIRELESS COMMUNICATIONS MAGAZINE during 2006–2010. She
is currently an Editor for the IEEE T RANSACTIONS ON WIRELESS COM-
MUNICATIONS, the IEEE TRANSACTIONS ON COMMUNICATIONS,andthe
IEEE COMMUNICATIONS MAGAZINE and is an Associate Editor for the Wiley
Security and Communication Networks Journal. She was the recipient of the
Quebec Government FQRNT Strategic Fellowship for Professors–Researchers
during 2001–2006; the INRS-EMT Performance Award in 2004 for outstanding
achievements in research, teaching, and service; the IEEE Communications
Society Certificate of Appreciation in 2006, 2009, and 2010; the Technical
Community Service Award from the FQRNT Center for Advanced Systems
and Technologies in Communications in 2007; and the Natural Sciences and
Engineering Research Council of Canada Discovery Accelerator Supplement
Award. She was also a corecipient of the Best Paper Award from the IEEE
Symposium on Computers and Communications in 2009, the International
Symposium on Wireless Personal Multimedia Communications in 2010, and
the IEEE Wireless Communications and Networking Conference in 2010.
